Cardinalities of $\mathbb{R}[X]$ and $\mathbb{R}$

Question

I want to show that the Cardinalities of $\mathbb{R}[X]$ and $\mathbb{R}$ are equal.

For starters, I tried mapping each polynomial to $\mathbb{R}^n$ by mapping it to its coordinates vector, but its not correct (I am limiting my polynomials here)

I can use cardinalities arithmetic and Cantor - Bernstein theorem.

Would love to get some insight.

Answer 1

$\mathbb R_n[X]$ the set of polynomials of degree $n$ has the same cardinal than $\mathbb R^{n+1}$, which has for cardinal the cardinal of $\mathbb R$. This is the consequence that for an infinite set $X$, the cardinality of $X \times X$ is the one of $X$.

Then $\mathbb R[X] = \bigcup_n \mathbb R_n[X]$ and therefore the cardinality of $\mathbb R[X]$ is equal to the one of $\mathbb R$.

Answer 2

The set of polynomials of degree $\le n$ is clearly in bijection with $\Bbb R^n$, which again is equinumerous to $\Bbb R$, or with the open interval $]n,n+1[$. Then $\Bbb R[X]$ can be injected into $\bigcup_n \left]n,n+1\right[\subseteq \Bbb R$.

Answer 3

Let $\mathbb{R}_n[X]$ be the set of real polynomials of degree at most $n$. It is clear that $|\mathbb{R}|\leq |\mathbb{R}_n[X]| \leq |\mathbb{R}^{n+1}|= |\mathbb{R}|$ and thus $|\mathbb{R}_n[X]| = |\mathbb{R}|$. We have

$$\mathbb{R}[X] = \bigcup_{n=0}^\infty \mathbb{R}_n[X]$$

and thus

$$|\mathbb{R}| \leq|\mathbb{R}[X]| \leq \sum_{n\in \mathbb{N}} |\mathbb{R}_n[X]| = \sum_{n \in \mathbb{N}}|\mathbb{R}| = \aleph_0 |\mathbb{R}| = |\mathbb{R}|$$

we deduce that $|\mathbb{R}[X]| = |\mathbb{R}|$.